---
title: "Global Lyapunov functions — paper by Alfarano et al. (2024)"
description: "The central innovation is \"backward\" data generation: the authors first sample random Lyapunov… 41 references, cited by 6 papers here."
authors: ["Alberto Alfarano", "François Charton", "Amaury Hayat"]
year: 2024
venue: "arXiv:2410.08304"
source: "https://arxiv.org/abs/2410.08304"
doi: "10.48550/arxiv.2410.08304"
arxiv: "2410.08304"
licence: "CC BY 4.0 (arXiv)"
licence_url: "https://creativecommons.org/licenses/by/4.0/"
summary_licence: "Elamur Cosmic License"
canonical: "https://research.elamur.ai/free/en/papers/alfarano-2024-global-lyapunov-functions-long-standing/"
---
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# Global Lyapunov functions: a long-standing open problem in mathematics, with symbolic transformers

**Authors:** Alberto Alfarano, [François Charton](https://md.research.elamur.ai/free/en/authors/francois-charton/) and [Amaury Hayat](https://md.research.elamur.ai/free/en/authors/amaury-hayat/) · **Year:** 2024 · **Venue:** [arXiv:2410.08304](https://md.research.elamur.ai/free/en/venues/neural-information-processing-systems/) ([NeurIPS 2024](https://md.research.elamur.ai/free/en/venues/neural-information-processing-systems/2024/)) · **Licence:** [CC BY 4.0 (arXiv)](https://creativecommons.org/licenses/by/4.0/)

**Links:** [Publisher's page](https://arxiv.org/abs/2410.08304) · [PDF](https://arxiv.org/pdf/2410.08304v1) · [DOI 10.48550/arxiv.2410.08304](https://doi.org/10.48550/arxiv.2410.08304) · [arXiv 2410.08304](https://arxiv.org/abs/2410.08304) · [OpenAlex](https://openalex.org/W4415796507)

**Citations:** 13 citations (OpenAlex, 2026-10-14) · [6](https://md.research.elamur.ai/free/en/papers/alfarano-2024-global-lyapunov-functions-long-standing/cited-by/) in our library, of the 149,439 papers whose bibliographies we read

## Summary
Proving that a dynamical system is globally stable usually means finding a Lyapunov function. No general recipe exists for this, and algorithmic tools such as SOSTOOLS only handle small polynomial systems. The authors treat the search as symbolic translation. A sequence-to-sequence transformer reads a system written as prefix-notation token sequences and outputs a candidate function. The central innovation is "backward" data generation: the authors first sample random Lyapunov functions, then build systems that those functions stabilize. This yields millions of training pairs for a problem where solving forward is generally infeasible. Mixing in a few hundred easier, algorithmically solved examples improves out-of-distribution generalization. The resulting models outperform SOS-based solvers and math students. They also find previously unknown Lyapunov functions for non-polynomial systems, a class with no known algorithm. Every output is an explicit formula that can be checked rigorously. The work therefore offers a template for using generative models to propose verifiable candidate solutions to open mathematical problems.

## Key points
- **Architecture and training:** 8 layers, 10 heads, embedding size 640, trained on about 2.4M examples per epoch with 8 V100 GPUs.
- **Accuracy:** about 99% on held-out in-distribution tests and 73% out-of-distribution. Adding 300 easy forward-generated examples raises out-of-distribution accuracy to 84%.
- **Random polynomial systems:** the models find Lyapunov functions for 10.1% of systems, compared with 2.1% for SOS-based methods.
- **Random non-polynomial systems:** the best model discovers new Lyapunov functions for 12.7% of systems. Adding 1,000 verified predictions back into training (expert iteration) improves these results by roughly 15%.
- **Human comparison and limits:** 25 Master's students solved 9.33% of problems drawn from FSOSTOOLS. The studied systems are small: polynomial systems have no more than 5 equations, and non-polynomial systems no more than 3.

*Our summary of the paper, in our own words — not its text. The work belongs to its authors; read it at the links above.*

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## Cite

Cite as: Alberto Alfarano, François Charton and Amaury Hayat. 2024. Global Lyapunov functions: a long-standing open problem in mathematics, with symbolic transformers. arXiv:2410.08304.

The paper itself, not this page. BibTeX from the library's record of the work.

```bibtex
@misc{alfarano2024global,
  title = {{Global Lyapunov functions: a long-standing open problem in mathematics, with symbolic transformers}},
  author = {Alberto Alfarano, François Charton and Amaury Hayat},
  year = {2024},
  doi = {10.48550/arxiv.2410.08304},
  eprint = {2410.08304},
  archivePrefix = {arXiv},
  url = {https://doi.org/10.48550/arxiv.2410.08304}
}
```

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## References

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*Updated 3 November 2026*